Problem: Rewrite as a simplified fraction. ${1.8\overline{32} = {?}}$
Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 1832.3232...\\ 10x &= 18.3232...\end{align*}}$ Subtracting the two equations, we have: ${990x = 1814}$ ${x = \dfrac{1814}{990}} $ ${x = \dfrac{907}{495}} \text{or} {1 \dfrac{412}{495}}$